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Probing the Stability of Gravastars by Dropping Dust Shells Onto Them Merse E Gáspár, István Rácz

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Probing the Stability of Gravastars by Dropping Dust Shells Onto Them Merse E Gáspár, István Rácz Probing the stability of gravastars by dropping dust shells onto them Merse E Gáspár, István Rácz To cite this version: Merse E Gáspár, István Rácz. Probing the stability of gravastars by dropping dust shells onto them. Classical and Quantum Gravity, IOP Publishing, 2010, 27 (18), pp.185004. 10.1088/0264- 9381/27/18/185004. hal-00625160 HAL Id: hal-00625160 https://hal.archives-ouvertes.fr/hal-00625160 Submitted on 21 Sep 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Probing the stability of gravastars by dropping dust shells onto them Merse E. G´asp´ar and Istv´an R´acz RMKI, H-1121 Budapest, Konkoly Thege Mikl´os ´ut 29-33. Hungary E-mail: [email protected], [email protected] Abstract. As a preparation for the dynamical investigations, this paper starts by providing a short review of the three-layer gravastar model with distinguished attention to the structure of the pertinent parameter space of gravastars in equilibrium. Then the radial stability of these type of gravastars is studied by determining their response ◦ for the totally inelastic collision of their surface layer with a dust shell. It is assumed ◦ that dominant energy condition holds and the speed of sound do not exceed that of ◦ the light in the matter of the surface layer. While in the analytic setup the equation of state is kept to be generic, in the numerical investigations three functionally distinct class of equation of states are applied. In the corresponding particular cases the maximal mass of the dust shell that may fall onto a gravastar without converting it into a black hole is determined. For those configurations which remain stable the excursion of their radius is assigned. It is found that even the most compact gravastars cannot get beyond the lower limit of the size of conventional stars, provided that the dominant energy condition holds in both cases. It is also shown—independent of any assumption concerning the matter interbridging the internal de Sitter and the external Schwarzschild regions—that the better is a gravastar in mimicing a black hole the easier is to get the system formed by a dust shell and the gravastar beyond itsthe event horizon of the composite system. In addition, a generic description of the totally ◦ inelastic collision of spherical shells in spherically symmetric spacetimes is also provided ◦ in the appendix. ◦ PACS numbers: 04.40.Dg, 97.60.Lf, 04.20.Jb, 95.36.+x Submitted to: Class. Quantum Grav. 1. Introduction There are more and more astrophysical observations justifying the existence of extremely compact massive objects with size close to their Schwarzschild radius [1, 2, 3]. It is widely held that these observations do also provide indirect justifications of the existence of black holes (BHs). Nevertheless, there are also alternative ideas around claiming that exotic states of matter may exist which could stabilize extremely compact stars suiting to the aforementioned astrophysical observations (see, e.g., [4] for a recent review). One of Probing the stability of gravastars by dropping dust shells onto them 2 the most popular among these type of BH mimicing objects is the gravitational vacuum star (gravastar) model which has received considerable attention not least because its relation to the concept of dark energy. In this model of Mazur and Mottola [5] an interior de Sitter spacetime region is connected via three intermediate layers to an outer Schwarzschild solution such that the radius of the outermost layer is supposed to be slightly larger than the Schwarzschild radius of the system. It is worth mentioning that in advance to the gravastar model there were several constructions in which the matching of a de Sitter region to the Schwarzschild spacetime was applied. For instance, to get read of the r = 0 singularity of the Schwarzschild spacetime Frolov, Markov and Mukhanov in [6] proposed a matching of a de Sitter interior to it at a small radius of Planck scale ensuring thereby that the curvature remains bounded everywhere in the yielded spacetime. Dynamical investigation of this model was already carried out in [7]. Note that up to certain extent the model used in [6] could be considered as the precursor of the gravastar model—although in the latter the matching was made in a more elaborated way—and the outermost matching surface, at the boundary of the Schwarzschild region, was supposed to be arranged such that its radius is slightly larger than the pertinent Schwarzschild radius. Once such a model is set up the following questions manifest themselves: (1) What type of physical process may produce such a gravastar? (2) Is a gravastar stable? (3) If it is, does it provide a viable alternative to BHs? While the first question has not been tackled yet even the second question turned to be too complex within the original model of Mazur and Mottola—although in [5] an argument claiming for the thermodynamical stability of it was given—as it is composed by making use of three different types of regions with unspecified matter. To reduce the related ambiguities Visser and Wiltshire [8] introduced a simplified three-layer gravastar model where the interior de Sitter region is matched to the exterior Schwarzschild spacetime via a single matter shell. This model is simple enough to carry out various analytic investigations by making use of the thin-shell formalism of Israel [9]. Visser and Wiltshire besides deriving the basic relations determining the evolution of gravastars did also carried out the first investigation of their radial stability. Since then the stability of gravastars has been studied by several authors within this simplified model or within its continuum correspondence [10]. Results relevant for radial stability may be found in [11, 12], and in case of electrically charged gravastars in [13]. The stability has also been investigated with respect to axial perturbations [14, 15]. In all of these investigations attention was restricted to the space of gravastars in equilibrium, i.e., the radial stability was investigated by determining the response of a gravastar to a slight formal change of the underlying effective potential. For instance, in [16, 17] the excursions of gravastars was investigated this way such that their evolution started with carefully prepared initial conditions. In all of the pertinent investigations it Probing the stability of gravastars by dropping dust shells onto them 3 was demonstrated that by suitably adjusting the equation of state (EOS) of the matter forming the surface of the gravastar the subspace of “stable” gravastars may always be ensured to be of non-zero measure. Nevertheless, it was also found that whenever the measure of the subspaces of the configuration space representing stable and unstable gravastars are compared the former is always found to be negligible with respect to the latter. This observation was commonly interpreted that gravastars may not offer a viable alternative to BHs. The main purpose of the present paper is to determine the response of a gravastar in equilibrium to the arrival of a dust shell onto its surface. This is done—not merely by considering some formal change of the effective potential determining the state of a gravastar—but by making use of the full dynamical setup. For the sake of definiteness, we assume that the surface of the gravastar and the dust shell collide in a totally inelastic manner. In addition, concrete EOSs are chosen and it is assumed that the dominant energy condition (DEC) holds and the speed of sound in the surface of the gravastar ‡ does not excide that of the light. Then the relevant non-linear problem—the basic equations of which are based on the generic dynamics of spherical shells—are solved by using analytic and numerical approaches. In this way not only the excursion of particular gravastar models may be studied but we could determine the maximal mass of the dust shell colliding with the surface of the gravastar without converting the latter into a BH. This paper is organized as follows. In Section 2 some of the basics of the Visser and Wiltshire three-layer dynamics gravastar model are recalled using dimensionless variables. As a byproduct of our preparation for the aforementioned dynamical investigations a short survey of the configuration space of stable gravastars is also ◦ ◦ provided . (Although there are no completely new results in this section we believe that this review provides a good reference frame for the results of the succeeding dynamical investigations.) In Section 3 the dynamics of the system composed by the spherically symmetric dust shell falling onto a stable gravastar, along with their ◦ collision, is described. Section 4 is to report about our analytic and numerical results concerning the dynamics of maximally loaded gravastars, while Section 5 contains the ◦ finalour concluding remarks. Finally, in the appendix, a generic description of the totally ◦ inelastic collision of spherical shells in spherically symmetric spacetimes is also provided. ◦ ◦ Throughout this paper the geometrized units, with G = c = 1, are applied. ◦ 2. Gravastar model of Visser and Wiltshire Throughout this paper considerations will be restricted to the three-layer spherically symmetric gravastar model of Visser and Wiltshire [8]. This simplified model consists of an external Schwarzschild vacuum region with mass parameter M—representing the Recall that the dominant energy condition guarantees that the concept of causality is properly ‡ adopted in general relativity.
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